antoine deza, mcmaster tamás terlaky, lehigh yuriy zinchenko, calgary feng xie, mcmaster
DESCRIPTION
Antoine Deza, McMaster Tamás Terlaky, Lehigh Yuriy Zinchenko, Calgary Feng Xie, McMaster. Hyperplane arrangements with large average diameter. P 2. P 1. P 4. P 6. P 5. P 3. Polytopes & Arrangements : Diameter & Curvature. Motivations and algorithmic issues. - PowerPoint PPT PresentationTRANSCRIPT
Antoine Deza, McMasterTamás Terlaky, Lehigh
Yuriy Zinchenko, CalgaryFeng Xie, McMaster
Hyperplane arrangements with large average diameter
P1
P2
P3
P4
P6 P5
Polytopes & Arrangements : Diameter & Curvature
Motivations and algorithmic issues
Diameter (of a polytope) :
lower bound for the number of iterationsfor the simplex method (pivoting methods)
Curvature (of the central path associated to a polytope) :
large curvature indicates large number of iterationsfor (path following) interior point methods
Polytopes & Arrangements : Diameter & Curvature
Polytope P defined by n inequalities in dimension d
n = 5 : inequalities d = 2 : dimension
polytope : bounded polyhedron
Polytopes & Arrangements : Diameter & Curvature
Polytope P defined by n inequalities in dimension d
n = 5 : inequalities d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
P
Polytope P defined by n inequalities in dimension d
n = 5 : inequalities d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
vertex c1
vertex c2
Diameter (P): smallest number such that any two vertices (c1,c2) can be connected by a path with at most (P) edges
P
n = 5 : inequalities d = 2 : dimension(P) = 2 : diameter
Polytopes & Arrangements : Diameter & Curvature
Diameter (P): smallest number such that any two vertices can be connected by a path with at most (P) edges
Hirsch Conjecture (1957): (P) ≤ n - d
P
n = 5 : inequalities d = 2 : dimension(P) = 2 : diameter
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
c
P
n = 5 : inequalities d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
c
P
n = 5 : inequalities d = 2 : dimension
c(P): redundant inequalities count
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
(P): largest total curvature c(P) over of all possible c
c
P
n = 5 : inequalities d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
(P): largest total curvature c(P) over of all possible c
Continuous analogue of Hirsch Conjecture: (P) = O(n)
c
P
n = 5 : inequalities d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
(P): largest total curvature c(P) over of all possible c
Continuous analogue of Hirsch Conjecture: (P) = O(n)
c
P
n = 5 : inequalities d = 2 : dimension
Dedieu-Shub hypothesis (2005): (P) = O(d)
Polytopes & Arrangements : Diameter & Curvature
c(P): total curvature of the primal central path of maxcTx : xP
(P): largest total curvature c(P) over of all possible c
Continuous analogue of Hirsch Conjecture: (P) = O(n)
c
P
n = 5 : inequalities d = 2 : dimension
Dedieu-Shub hypothesis (2005): (P) = O(d) D.-T.-Z. (2008) polytope C such that: (C) ≥ (1.5)d
• start from the analytic center• follow the central path• converge to an optimal solution• polynomial time algorithms for linear optimization : number of iterations
n : number of inequalities L : input-data bit-length
( )O nL
analytic center
centralpathoptimal
solution
Polytopes & Arrangements : Diameter & CurvatureCentral path following interior point methods
: central path parameter xP : Ax ≥ b
max ln( )ii
x Ax b cc
total curvature
C2 curve : [a, b] Rd parameterized by its arc length t (note: (t) = 1)
curvature at t : )()( 2
2
tt
t
y = sin (x)
total curvature
Polytopes & Arrangements : Diameter & Curvature
C2 curve : [a, b] Rn parameterized by its arc length t (note: (t) = 1)
curvature at t :
total curvature :
total curvature
)()( 2
2
tt
t
b
a
dttK )(
y = sin (x)
total curvature
Polytopes & Arrangements : Diameter & Curvature
Polytopes & Arrangements : Diameter & Curvature
Arrangement A defined by n hyperplanes in dimension d
n = 5 : hyperplanes d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
Simple arrangement: n d and any d hyperplanes intersect at a unique distinct point
n = 5 : hyperplanes d = 2 : dimension
Polytopes & Arrangements : Diameter & Curvature
For a simple arrangement, the number of bounded cells I =1
nd
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
P1
P2
P3
P4
P6 P5
Polytopes & Arrangements : Diameter & Curvature
c(A) : average value of c(Pi) over the bounded cells Pi of A:
c(A) = with I =1
( )i
ii
P
I
c
I1
nd
c
P1
P3
P2
P6
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
c(Pi): redundant inequalities count
P5
P4
Polytopes & Arrangements : Diameter & Curvature
c(A) : average value of c(Pi) over the bounded cells Pi of A:
(A) : largest value of c(A) over all possible c
P1
P3
P2
P6
c
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
P5
P4
Polytopes & Arrangements : Diameter & Curvature
c(A) : average value of c(Pi) over the bounded cells Pi of A:
(A) : largest value of c(A) over all possible c
Dedieu-Malajovich-Shub (2005): (A) ≤ 2 d
P1
P3
P2
P6
c
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
P5
P4
Polytopes & Arrangements : Diameter & Curvature
(A) : average diameter of a bounded cell of A:
P1
P3
P2
P6
P4
P5
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
A : simple arrangement
Polytopes & Arrangements : Diameter & Curvature
(A) : average diameter of a bounded cell of A:
(A) = with I =1
( )i
ii
P
I
I
P1
P3
P2
P6
1
nd
P4
P5
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
(A): average diameter ≠ diameter of A ex: (A)= 1.333…
Polytopes & Arrangements : Diameter & Curvature
(A) : average diameter of a bounded cell of A:
(A) = with I =1
( )i
ii
P
I
I
P1
P3
P2
P6
1
nd
P4
P5
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
(Pi): only active inequalities count
Polytopes & Arrangements : Diameter & Curvature
P1
P3
P2
P6
P4
P5
n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells
(A) : average diameter of a bounded cell of A:
Conjecture [D.-T.-Z.]: (A) ≤ d (discrete analogue of Dedieu-Malajovich-Shub result)
Polytopes & Arrangements : Diameter & Curvature
(P) ≤ n - d (Hirsch conjecture 1957)
(A) ≤ d (conjecture D.-T.-Z. 2008) (A) ≤ 2 d (Dedieu-Malajovich-Shub 2005)
(P) ≤ 2 n (conjecture D.-T.-Z. 2008)
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
• for unbounded polyhedra• Hirsch conjecture is not valid• central path may not be defined
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
• for unbounded polyhedra• Hirsch conjecture is not valid• central path may not be defined
• Hirsch conjecture (P) ≤ n - d implies that (A) ≤ d
• Hirsch conjecture holds for d = 2, (A) ≤ 2
• Hirsch conjecture holds for d = 3, (A) ≤ 3
• Barnette (1974) and Kalai-Kleitman (1992) bounds imply
• (A) ≤
• (A) ≤
11
nn
1nn
11
nn
log2
4 21
1
dn
nd ndd
n
21 5 21 2 3
dndn d
• bounded cells of A* : ( n – 2) (n – 1) / 2 4-gons and n - 2 triangles
n = 6 d = 2
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension 2
(A) ≥ 2 -
2 / 2( 1)( 2)
n
n n
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension 2
(A) = 2 -
maximize(diameter) amounts tominimize( #external edge + #odd-gons)
2 / 2( 1)( 2)
n
n n
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension 3
(A) ≤ 3 +
and A* satisfies
(A*) = 3 -
24(2 16 21)3( 1)( 2)( 3)
n n
n n n
6 / 2 261 ( 1)( 2)( 3)
nn n n n
A*
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension d
A* satisfies
(A*) ≥ d
A* cyclic arrangement
- -1
/n d nd d
A* mainly consists of cubical cells
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension d
A* satisfies
(A*) ≥ n
Haimovich’s probabilistic analysis of shadow-vertex simplex method - Borgwardt (1987) Ziegler (1992) combinatorics of (pseudo) hyperplane addition to A* (Bruhat order) Forge – Ramírez Alfonsín (2001) counting k-face cells of A*
- -1
/n d nd d
Polytopes & Arrangements : Diameter & CurvatureLinks, low dimensions and substantiation
dimension 2 (A) =
dimension 3 (A) asympotically equal to 3 dimension d d ≤ (A) ≤ d ?
[D.-Xie 2008]
- -1
/n d nd d
Haimovich’s probabilistic analysis of shadow-vertex simplex method - Borgwardt (1987) Ziegler (1992) combinatorics of (pseudo) hyperplane addition to A* (Bruhat order) Forge – Ramírez Alfonsín (2001) counting k-face cells of A*
2 / 2( 1)( 2)
n
n n
Polytopes & Arrangements : Diameter & Curvature
bounded / unbounded cells (sphere arrangements)?
simple / degenerate arrangements? pseudo-hyperplanes / underlying oriented matroid ?
Is it the right setting?(forget optimization and the vector c)
Polytopes & Arrangements : Diameter & Curvature
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998)
Fritzsche-Holt (1999) Holt (2004)
Links, low dimensions and substantiation
Polytopes & Arrangements : Diameter & Curvature
Links, low dimensions and substantiation
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998)
Fritzsche-Holt (1999) Holt (2004)
Is the continuous analogue true?
Polytopes & Arrangements : Diameter & Curvature
Links, low dimensions and substantiation
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998)
Fritzsche-Holt (1999) Holt (2004)
Is the continuous analogue true? yes
Polytopes & Arrangements : Diameter & Curvature
Links, low dimensions and substantiation
( ) such that lim inf
c
nP
nP
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998)
Fritzsche-Holt (1999) Holt (2004)
•continuous analogue of tightness:• for n ≥ d ≥ 2 D.-T.-Z. (2008)
Polytopes & Arrangements : Diameter & Curvature
slope: careful and geometric decrease
( ) such that liminf
c
nP
nP
Polytopes & Arrangements : Diameter & Curvature
slope: careful and geometric decrease
( ) such that liminf
c
nP
nP
Polytopes & Arrangements : Diameter & Curvature
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d
• continuous analogue of tightness:
• for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008)
Links, low dimensions and substantiation
Polytopes & Arrangements : Diameter & Curvature
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d
• continuous analogue of tightness:• for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008)
• Hirsch special case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967)
Links, low dimensions and substantiation
Polytopes & Arrangements : Diameter & Curvature
Links, low dimensions and substantiation
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d
• continuous analogue of tightness:• for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008)
• Hirsch special case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967)
Is the continuous analogue true?
Polytopes & Arrangements : Diameter & Curvature
Links, low dimensions and substantiation
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d
• continuous analogue of tightness:• for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008)
• Hirsch special case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967)
Is the continuous analogue true? yes
Polytopes & Arrangements : Diameter & Curvature
Links, low dimensions and substantiation
• Hirsch conjecture is tight • for n > d ≥ 7 P such that (P) ≥ n - d
• continuous analogue of tightness:• for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008)
• Hirsch special case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967)
•continuous analogue of d-step equivalence:• (P) = O(n) (P) = O(d) for n = 2d D.-T.-Z. (2008)
Polytopes & Arrangements : Diameter & Curvature
(P) ≤ n - d Hirsch conjecture (1957) (P) < n – d Holt-Klee (1998)
(A) ≤ d conjecture D.-T.-Z. (2008) (A) ≤ 2 d Dedieu-Malajovich-Shub (2005)
(P) ≤ 2 n conjecture D.-T.-Z. (2008) (P)=Ω(n) D.-T.-Z (2008)
holds for n = 2 and n = 3
holds for n = 2 and n = 3
(P) ≤ n – d (P) ≤ d for n = 2d Klee-Walkup (1967)(P) = O(n) (P) = O(d) for n = 2d D.-T.-Z. (2008)
redundant inequalities counts for (P)
Polytopes & Arrangements : Diameter & Curvature
(P) ≤ n - d Hirsch conjecture (1957) (P) < n – d Holt-Klee (1998)
(A) ≤ d conjecture D.-T.-Z. (2008) (A) ≤ 2 d Dedieu-Malajovich-Shub (2005)
(P) ≤ 2 n conjecture D.-T.-Z. (2008) (P)=Ω(n) D.-T.-Z (2008)
holds for n = 2 and n = 3
holds for n = 2 and n = 3
(P) ≤ n – d (P) ≤ d for n = 2d Klee-Walkup (1967)(P) = O(n) (P) = O(d) for n = 2d D.-T.-Z. (2008)
redundant inequalities counts for (P)
c
Want: (P) = O(n), n (P) = O(d) for n = 2d, d
Clearly (P) K n, n (P) K n for n = 2d, d
Need (P) K n, n (P) n for n = 2d, d
Case (i) d < n < 2d
e.g., P with n=3, d=2
P P with n=4, d=2 c(P) c(P) 2 d< 2 n
Polytopes & Arrangements : Diameter & CurvatureProof of a continuous analogue of Klee-Walkup result (d-step conjecture)
A4
Want: (P) = O(n), n (P) = O(d) for n = 2d, d
Need (P) K n, n (P) n for n = 2d, d
Case (ii) 2d < n
e.g., P with n=5, d=2
P P with n=6, d=3 c(P) (P) 2 (d + 1) … < 2 (n - d) < 2 n
Polytopes & Arrangements : Diameter & Curvature
c
c
Proof of a continuous analogue of Klee-Walkup result (d-step conjecture)
Polytopes & Arrangements : Diameter & Curvature
Motivations, algorithmic issues and analogies
Diameter (of a polytope) :
lower bound for the number of iterationsfor the simplex method (pivoting methods)
Curvature (of the central path associated to a polytope) :
large curvature indicates large number of iterationfor (central path following) interior point methods
we believe:
polynomial upper bound for (P) ≤ d (n – d) / 2
Polytopes & Arrangements : Diameter & Curvature
bounded / unbounded cells (sphere arrangements)?
simple / degenerate arrangements? pseudo-hyperplanes / underlying oriented matroid ?
Is it the right setting?(forget optimization and the vector c)
thank you
Polytopes & Arrangements : Diameter & Curvature
Motivations, algorithmic issues and analogies
Diameter (of a polytope) :
lower bound for the number of iterationsfor the simplex method (pivoting methods)
Curvature (of the central path associated to a polytope) :
large curvature indicates large number of iterationfor (central path following) interior point methods
we believe:
polynomial upper bound for (P) ≤ d (n – d) / 2
thank you